Question

1) At the following network, 900 cars are going from node to node 6 per hour. The values on arcs are maximum number of cars that can use the arc per hour and average time to traverse the are respectively. Find minimum total time for the travel of 900 cars by using capacitated network simplex method. 2-600, 30-4 800, 10 100,70 400, 60 6 )-900 600, 30 900 300, 10 600, 30 600, 50 3-400, 605

Answer 1

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Create Excel as shown below

fx SUMPRODUCT(C2:C10,E2:E10) J10 From To Time Capacity Flow Node Netflow RHS 2 10 900 50 600 2 30 80 100 4 70 300 80 10 400 60 -900 30 400 60 Minimum Time 0 30 10 600

FORMULAS

H2 = SUMIF ($A$2:$A$10,G2,$E$2:$E$10) - SUMIF ($B$2:$B$10,G2,$E$2:$E$10) copy to H2:H7

J10 = SUMPRODUCT (C2:C10,E2:E10)

Enter Solver parameters

S Solver Parameters Set Objective: SUS10 O Max ⓔMin O Value Of: To: By Changing Variable Cells: SES2:SES10 Subject to the Constraints: SES2:SE$10 <= SDS2:SDS10 Add Change Delete Reset AlI Make Unconstrained Variables Non-Negative Simplex LP Select a Solving Method: Options Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems Solve Close

Click Solve button to generate the optimal solution:

Node NetflowRHS From To Time Capacity Flow Solver Results 10 800 700 900 900 200 Solver found a solution All condtraints and optimality conditions are satistied 30 600 600 100 100 Keap Solver Solution 10 200 900 900 ORestore Original Value 400 Return to Solver Parameters Dialog 400 500Minimum Tme 95000 Minimum Time95000 Solver found a solution. All constraints and optimality conditions are satisfied. When Simplex LP is used, this means Solver has found a global optimal solution. 13 15

Optimal number of cars that should flow from each node to each node is in column E

Minimum total time = 95,000

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